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Group Velocity

Group Velocity :- When a single sinusoidal wave travels, it has one frequency and one wavelength, and it stretches infinitely in space. But real waves (like light pulses, water wave bursts, or matter waves) usually travel as wave packets, formed by the superposition of many waves of slightly different frequencies.

Such a packet has :

A fast oscillating inner pattern → carrier wave
A slowly moving envelope → wave group

The speed of this envelope is called the group velocity.

Definition

(Group Velocity)

Group velocity is the speed with which energy or information travels when a wave packet is formed by the superposition of many waves with slightly different wavelengths. In a dispersive medium, different wavelengths move with different phase velocities, so the energy does not travel with the phase velocity. Instead, it moves with the group velocity. Group velocity is the speed with which the overall shape (envelope) of a group of waves travels.

$\displaystyle {{\upsilon }_{g}}=\frac{d\omega }{dk}=\upsilon -\lambda \frac{d\upsilon }{d\lambda }$ …..(1)

where

ω = 2πν = angular frequency

k = 2π/λ = wave number

v = ω/k = wave velocity/phase velocity

Formation of a Wave Group

Consider two waves of nearly equal frequencies and wave numbers :

$\displaystyle {{y}_{1}}=Asin({{\omega }_{1}}t-{{k}_{1}}x)$ …..(2)

$\displaystyle {{y}_{2}}=Asin({{\omega }_{2}}t-{{k}_{2}}x)$ …..(3)

Adding,

$\displaystyle y={{y}_{1}}+{{y}_{2}}$

Using trigonometric identity :

$\displaystyle sinA+sinB=2sin\frac{A+B}{2}\text{ }cos\frac{A-B}{2}$

We get,

$\displaystyle y=2Asin\left[ \left( \frac{{{\omega }_{1}}+{{\omega }_{2}}}{2} \right)t-\left( \frac{{{k}_{1}}+{{k}_{2}}}{2} \right)x \right]cos\left[ \left( \frac{{{\omega }_{1}}-{{\omega }_{2}}}{2} \right)t-\left( \frac{{{k}_{1}}-{{k}_{2}}}{2} \right)x \right]$ …..(4)

Physical Meaning

The equation (4) has two parts :

1. Carrier Wave (fast oscillation)

$\displaystyle sin(\overline{\omega }t-\overline{k}x)$

This moves with phase velocity/wave velocity :

$\displaystyle \upsilon =\frac{\omega }{k}$

2. Envelope (slow modulation)

$\displaystyle cos\left( \frac{\Delta \omega }{2}t-\frac{\Delta k}{2}x \right)$

Getting the group velocity :

To track the motion of the envelope peak or to find the group velocity,

$\displaystyle \frac{\Delta \omega }{2}t-\frac{\Delta k}{2}x=constant$

Differentiate wrt time :

$\displaystyle \Delta \omega -\Delta k\frac{dx}{dt}=0$

$\displaystyle \Rightarrow \frac{dx}{dt}=\frac{\Delta \omega }{\Delta k}$

But a real wave packet contains many waves, with frequencies very close together. So we take the limit :

$\displaystyle {{\upsilon }_{g}}=\underset{\delta k\to 0}{\mathop{lim}}\,\frac{\Delta \omega }{\Delta k}=\frac{d\omega }{dk}$ …..(5)

That’s the group velocity.

Also, we know that the speed of a single wave travelling in a medium is given by

$\displaystyle \upsilon =\frac{\omega }{k}\Rightarrow \omega =k\upsilon$

Using equation (5),

$\displaystyle {{\upsilon }_{g}}=\frac{d\omega }{dk}=\frac{d}{dk}\left( k\upsilon \right)=\upsilon +k\frac{d\upsilon }{dk}$ …..(6)

Now as

$\displaystyle k=\frac{2\pi }{\lambda }$

$\displaystyle \Rightarrow \frac{dk}{d\lambda }=-\frac{2\pi }{{{\lambda }^{2}}}$

$\displaystyle \Rightarrow dk=-\frac{2\pi }{{{\lambda }^{2}}}d\lambda$

Using this in equation (6), we get

$\displaystyle {{\upsilon }_{g}}=\upsilon +k\frac{d\upsilon }{\left( -\frac{2\pi }{{{\lambda }^{2}}}d\lambda \right)}=\upsilon +\frac{2\pi }{\lambda }\frac{d\upsilon }{\left( -\frac{2\pi }{{{\lambda }^{2}}}d\lambda \right)}$

$\displaystyle \therefore {{\upsilon }_{g}}=\upsilon -\lambda \frac{d\upsilon }{d\lambda }$ …..(7)

✨ Special Case

If the medium is non-dispersive :

$\displaystyle \omega =\upsilon k$

$\displaystyle {{\upsilon }_{g}}=\frac{d\omega }{dk}=\frac{d}{dk}\left( \upsilon k \right)=\upsilon$

So group velocity = phase velocity.

But in dispersive media (water waves, light in glass, matter waves) :

$\displaystyle {{\upsilon }_{g}}\ne {{\upsilon }_{p}}$

🌊 What is Dispersion ?

Dispersion means : the wave speed depends on frequency (or wavelength). Mathematically, it means the relation between ω and k is not linear.

✅ Non-dispersive medium

A medium is non-dispersive if all frequencies travel with the same speed.

$\displaystyle \upsilon =constant$

Example 1.

In a certain dispersive medium, the phase velocity of waves varies with wavelength according to

$\displaystyle \upsilon =6\times {{10}^{8}}-2\times {{10}^{14}}\lambda$

where is in m/s and is in meters. Find the group velocity of a wave whose wavelength is λ = 4 × 10^-7 m.

Solution :

Using

$\displaystyle {{\upsilon }_{g}}=\upsilon -\lambda \frac{d\upsilon }{d\lambda }$

$\displaystyle {{\upsilon }_{g}}=\left[ 6\times {{10}^{8}}-2\times {{10}^{14}}\left( 4\times {{10}^{-7}} \right) \right]-\left( 4\times {{10}^{-7}} \right)\frac{d}{d\lambda }\left[ 6\times {{10}^{8}}-2\times {{10}^{14}}\lambda \right]$

$\displaystyle {{\upsilon }_{g}}=\left[ 6\times {{10}^{8}}-0.8\times {{10}^{8}} \right]-\left( 4\times {{10}^{-7}} \right)\times \left[ 0-2\times {{10}^{14}} \right]$